
Algorithmic Aspects of 2Secure Domination in Graphs
Let G(V,E) be a simple, undirected and connected graph. A dominating set...
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Sequences of radius k for complete bipartite graphs
A kradius sequence for a graph G is a sequence of vertices of G (typica...
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Shortest Reconfiguration of Matchings
Imagine that unlabelled tokens are placed on the edges of a graph, such ...
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Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph
A standard way to approximate the distance between any two vertices p an...
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Small World Model based on a Sphere Homeomorphic Geometry
We define a small world model over the octahedron surface and relate its...
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Minimum Scan Cover with Angular Transition Costs
We provide a comprehensive study of a natural geometric optimization pro...
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Efficient Reassembling of ThreeRegular Planar Graphs
A reassembling of a simple graph G = (V,E) is an abstraction of a proble...
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Monitoring the edges of a graph using distances
We introduce a new graphtheoretic concept in the area of network monitoring. A set M of vertices of a graph G is a distanceedgemonitoring set if for every edge e of G, there is a vertex x of M and a vertex y of G such that e belongs to all shortest paths between x and y. We denote by dem(G) the smallest size of such a set in G. The vertices of M represent distance probes in a network modeled by G; when the edge e fails, the distance from x to y increases, and thus we are able to detect the failure. It turns out that not only we can detect it, but we can even correctly locate the failing edge. In this paper, we initiate the study of this new concept. We show that for a nontrivial connected graph G of order n, 1≤ dem(G)≤ n1 with dem(G)=1 if and only if G is a tree, and dem(G)=n1 if and only if it is a complete graph. We compute the exact value of dem for grids, hypercubes, and complete bipartite graphs. Then, we relate dem to other standard graph parameters. We show that demG) is lowerbounded by the arboricity of the graph, and upperbounded by its vertex cover number. It is also upperbounded by twice its feedback edge set number. Moreover, we characterize connected graphs G with dem(G)=2. Then, we show that determining dem(G) for an input graph G is an NPcomplete problem, even for apex graphs. There exists a polynomialtime logarithmicfactor approximation algorithm, however it is NPhard to compute an asymptotically better approximation, even for bipartite graphs of small diameter and for bipartite subcubic graphs. For such instances, the problem is also unlikey to be fixed parameter tractable when parameterized by the solution size.
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